Structural Reliability Applications of Nonstationary Spectral Characteristics

Abstract

This paper presents new closed-form analytical approximations to the first-passage problem in structural reliability by using the exact closed-form solutions for the spectral characteristics of nonstationary random processes. The first-passage problem applied to a structural system possibly with random parameters and subjected to stochastic loading consists of computing the probability of a response quantity exceeding a deterministic threshold in a given exposure time. This paper also investigates, on the basis of benchmark problems, the absolute and relative accuracy of analytical approximations of the time-variant failure probability, such as Poisson, classical Vanmarcke, and modified Vanmarcke approximations, in the case of nonstationary random vibration. The classical and modified Vanmarcke approximations are expressed as time integrals of the closed forms of the corresponding hazard functions. These closed forms refer to linear elastic systems subjected to stationary and nonstationary base excitation from at-rest initial conditions, and they are obtained using recently developed exact closed-form solutions for the time-variant bandwidth parameter. These closed-form Vanmarcke’s approximate solutions to the first-passage problem are compared with the well known Poisson approximation and accurate simulation results obtained via the importance sampling using elementary events (ISEE) method for two benchmark applications: (1) a set of linear elastic single-degree-of-freedom (SDOF) systems defined by different natural periods and damping ratios, and (2) an idealized, yet realistic three-dimensional asymmetric steel building model. The linear elastic SDOF systems are subjected to white noise base excitation from at-rest initial conditions, while the steel building model is subjected, from at-rest initial conditions, first to white noise and then to a time-modulated colored noise base excitation. The retrofit of this second benchmark structure with viscous dampers is also considered, illustrating (1) the use of the newly available closed-form approximations of the failure probability for nonclassically damped linear elastic systems, and (2) an example of practical use in structural engineering of the presented analytical solutions. The results presented in this study show that, for nonstationary random vibration problems, the two Vanmarcke approximations can improve considerably the estimates of the time-variant failure probability for the first-passage problem when compared with the simpler Poisson approximation.